# Lines, Angles and Pairs of Lines - Link: https://thecuriosi.com/lessons/aptet-maths-lines-angles/ - Published: 2026-11-03T18:29:45+05:30 - Author: MAHENDRA MANCHIKALAPATI 1. [Courses](https://thecuriosi.com/courses/) 2. [APTET Maths: Geometry, Algebra, Data and Methodology](https://thecuriosi.com/courses/aptet-maths-geometry-algebra-methodology/) 3. Lesson 1 of 10 Geometry begins with a few simple ideas: points, lines and angles. The APTET syllabus lists **point, line, line segment, ray, curves, polygons, angles**, **measuring lines**, **pairs of lines**, and the **elements, measurement and types of angles**. These terms are the vocabulary of every later geometry question, so precise definitions matter. ## Basic geometrical ideas | Term | Meaning | Key fact | | Point | an exact position; it has no length, breadth or thickness | named by a capital letter: point A | | Line | a straight path extending endlessly in both directions | **no end points**; cannot be measured | | Line segment | part of a line between two points | **two end points**; has a definite length | | Ray | part of a line starting at one point and going endlessly in one direction | **one end point** (the initial point) | | Collinear points | points lying on the same line | three or more points may or may not be collinear | | Plane | a flat surface extending endlessly | a sheet of paper is a model of part of a plane | Through **one point** infinitely many lines can be drawn; through **two distinct points** exactly **one** line can be drawn. **Curves:** any drawing made without lifting the pencil is a curve (in mathematics, a straight line is also a curve). A **simple curve** does not cross itself; a ** closed curve** ends where it starts; an **open curve** does not. A **polygon** is a simple closed figure made only of line segments (triangle, quadrilateral, pentagon…). **Measuring a line segment:** use a ruler (or a divider and ruler). Place the zero mark at one end, or measure between two marks and subtract. Keep the eye **vertically above** the mark to avoid **parallax error**. Units: 10 mm = 1 cm. ## Angles: elements, measurement and types An **angle** is formed by two rays with a common end point. The common point is the **vertex** and the rays are the **arms**. ∠ABC has vertex B. Angles are measured in **degrees** with a **protractor**; a complete turn is 360°. | Type | Measure | Example | | Zero angle | 0° | both arms together | | Acute | between 0° and 90° | 45° | | Right | exactly 90° | corner of a page | | Obtuse | between 90° and 180° | 120° | | Straight | exactly 180° | a straight line | | Reflex | between 180° and 360° | 210° | | Complete | 360° | one full turn | **Clock angles:** the hour hand covers 30° each hour (360° ÷ 12). At 3:00 the angle is 90°; at 4:00 it is 4 × 30° = **120°**. ## Pairs of angles and pairs of lines | Pair | Rule | | Complementary angles | sum is 90° (complement of 35° is 55°) | | Supplementary angles | sum is 180° (supplement of 65° is 115°) | | Adjacent angles | common vertex, common arm, no common interior | | Linear pair | adjacent angles whose non-common arms form a straight line; sum is 180° | | Vertically opposite angles | formed when two lines intersect; they are **equal** | **Worked example:** two angles form a linear pair and one is 4 times the other. x + 4x = 180°, so 5x = 180° and x = **36°**; the angles are 36° and 144°. ### Pairs of lines * **Intersecting lines** meet at one point. * **Parallel lines** lie in the same plane and never meet (railway tracks); written l ∥ m. * **Perpendicular lines** meet at a right angle; written l ⊥ m. A **transversal** is a line that cuts two or more lines at different points. When it cuts **two parallel lines**: | Angle pair | Relation | | Corresponding angles | equal | | Alternate interior angles | equal | | Alternate exterior angles | equal | | Interior angles on the same side (co-interior) | supplementary (sum 180°) | So if one co-interior angle is 70°, the other is 180° − 70° = **110°**. ### Exam focus * Line: no end points; ray: one; segment: two. * Reflex angles lie between 180° and 360°. * Complementary = 90°, supplementary = 180°; vertically opposite angles are equal. * Parallel lines and a transversal: corresponding and alternate angles equal; co- interior angles add to 180°. * Clock: 30° per hour mark. ### Example Mrs Hemalatha gives Class 4 straws and pins. Children make angles, compare them with the corner of a notebook (a right angle) and sort them into “smaller than a corner” (acute) and “bigger than a corner” (obtuse). Only after this do they measure with a protractor. ## Key takeaways * A point has position only; a line has no end points; a segment has two; a ray has one. * Exactly one line passes through two distinct points. * Angles are acute, right, obtuse, straight, reflex or complete. * Complementary angles total 90°; supplementary angles and linear pairs total 180 °. * Vertically opposite angles are equal. * With parallel lines, corresponding and alternate angles are equal, co-interior angles are supplementary. ## Practice 1. A line segment has: 2. * (A) no end points * (B) one end point * (C) two end points * (D) three end points 3. The complement of 35° is: 4. * (A) 55° * (B) 145° * (C) 65° * (D) 325° 5. An angle of 210° is: 6. * (A) an obtuse angle * (B) a straight angle * (C) an acute angle * (D) a reflex angle 7. How many lines can pass through two distinct points? 8. * (A) infinitely many * (B) exactly one * (C) two * (D) none 9. The angle between the hands of a clock at 4:00 is: 10. * (A) 100° * (B) 120° * (C) 140° * (D) 90° 11. Two parallel lines are cut by a transversal. One interior angle on the same side of the transversal is 70°. The other is: 12. * (A) 70° * (B) 20° * (C) 90° * (D) 110° 13. Two lines intersect and one of the angles formed is 50°. The vertically opposite angle is: 14. * (A) 50° * (B) 130° * (C) 40° * (D) 310° 15. Two angles form a linear pair and one is four times the other. The smaller angle is: 16. * (A) 45° * (B) 30° * (C) 36° * (D) 144° Answers and explanations (tap to show) 1. (C) A line segment has a definite length between two end points. 2. (A) 90° − 35° = 55°. 3. (D) 210° lies between 180° and 360°. 4. (B) Exactly one line passes through two distinct points. 5. (B) 4 hour marks × 30° = 120°. 6. (D) Co-interior angles are supplementary: 180° − 70° = 110°. 7. (A) Vertically opposite angles are equal. 8. (C) x + 4x = 180°, so x = 36°. 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